Physics | Wave phenomena

Hello Future Physicists! Welcome to Wave Phenomena

Welcome to the exciting world of Wave Phenomena! If you’ve ever wondered how your phone receives Wi-Fi signals, why the sound of an ambulance changes pitch as it passes, or how musical instruments produce perfect notes, this chapter holds the answers.

This section is all about what happens when waves—whether they are light, sound, or ripples in water—meet obstacles, change mediums, or crash into each other. It’s a core component of Physics that explains almost everything we experience in terms of communication and perception.

Don't worry if some concepts seem tricky at first. We will break them down using everyday analogies to make sure every concept sticks!

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C.3 Wave Interactions: Reflection, Refraction, and Diffraction

1. Reflection (Bouncing Back)

Reflection is the phenomenon where a wave changes direction at a boundary between two different media, staying in the original medium.

Key Principle: The Law of Reflection

• The angle of incidence (\(\theta_i\)) is the angle between the incident ray and the normal (a line perpendicular to the surface).
• The angle of reflection (\(\theta_r\)) is the angle between the reflected ray and the normal.

The law is simple: The angle of incidence equals the angle of reflection.

\[ \theta_i = \theta_r \]

Example: This is why mirrors work! The light waves hit the smooth surface and reflect back to your eyes at the exact same angle they came in at.

2. Refraction (Bending and Changing Speed)

Refraction is the change in direction of a wave as it passes from one medium to another (e.g., from air to water), caused by a change in the wave's speed.

The Analogy of the Car in the Mud

Imagine driving a car from pavement (fast medium) onto a patch of mud (slow medium) at an angle. The first wheel to hit the mud slows down, while the other wheel is still moving fast on the pavement. This difference in speed causes the car (the wave) to pivot and change direction (bend).

• When waves go from a fast medium to a slow medium (e.g., air to glass), they bend towards the normal.
• When waves go from a slow medium to a fast medium (e.g., glass to air), they bend away from the normal.

Key takeaway: Refraction happens because the wave speed (\(v\)) and the wavelength (\(\lambda\)) change, but the frequency (\(f\)) remains constant.

3. Diffraction (Spreading Out)

Diffraction is the spreading of waves as they pass through an opening (aperture) or move around an obstacle.

Example: You can hear someone talking around a corner even if you can't see them. The sound waves (which have a large wavelength) diffract easily around the wall.

The Diffraction Rule

The amount of spreading depends critically on the relationship between the wave's wavelength (\(\lambda\)) and the size of the gap or obstacle (\(d\)).

Maximum Diffraction: Occurs when the wavelength is approximately equal to the size of the gap or aperture (\(\lambda \approx d\)).

Minimal Diffraction: Occurs when the wavelength is much smaller than the gap (\(\lambda \ll d\)). This is why light (very small \(\lambda\)) doesn't diffract noticeably around everyday objects unless the gap is extremely small (like in a double-slit experiment).

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C.3 The Principle of Superposition and Interference

1. The Principle of Superposition

When two or more waves overlap in the same region of space, the Principle of Superposition states that the resultant displacement at any point is the vector sum of the displacements of the individual waves at that point.

In simple terms: You just add the waves together!

2. Interference

The result of the superposition of two waves is called interference. There are two main types:

A. Constructive Interference

Occurs when waves meet in phase (crest meets crest, or trough meets trough). They reinforce each other, resulting in a resultant wave with a larger amplitude.

• Condition for Constructive Interference: The path difference must be zero or an integer multiple of the wavelength (\(n\lambda\), where \(n = 0, 1, 2, 3...\)).

B. Destructive Interference

Occurs when waves meet out of phase (crest meets trough). They cancel each other out, resulting in a resultant wave with a smaller (or zero) amplitude.

• Condition for Destructive Interference: The path difference must be an odd integer multiple of half a wavelength (\((n + \frac{1}{2})\lambda\), where \(n = 0, 1, 2, 3...\)).

Important Concept: Coherence

For interference patterns (like bright and dark fringes for light, or loud and quiet spots for sound) to be stable and observable, the sources must be coherent.

Coherent sources are sources that have the same frequency and maintain a constant phase difference between them.

Did you know? Noise-cancelling headphones use destructive interference. They generate an inverted sound wave (180° out of phase) to cancel out incoming ambient noise, leaving you with silence or just your music!

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C.4 Standing Waves and Resonance (SL/HL)

1. Formation of Standing Waves

A standing wave (or stationary wave) is the result of the superposition of two identical waves traveling in opposite directions in the same medium. Unlike traveling waves, energy is not propagated through the medium; it remains contained between the nodes.

Key Features of a Standing Wave

• Nodes (N): Points along the standing wave that always have zero displacement. These points do not move.
• Antinodes (A): Points along the standing wave that experience maximum displacement (maximum amplitude).

The distance between two consecutive nodes (or two consecutive antinodes) is always \(\frac{1}{2}\lambda\).
The distance between a node and the next antinode is \(\frac{1}{4}\lambda\).

2. Harmonics and Boundary Conditions

Standing waves only form when specific conditions are met, usually related to the length (\(L\)) of the medium. These specific patterns are called harmonics, or resonant frequencies.

A. Standing Waves in Strings (Fixed at both ends)

Since the ends are fixed, they must always be Nodes.

• First Harmonic (Fundamental Frequency, \(f_1\)): The simplest pattern, consisting of one antinode. The length \(L\) is equal to half a wavelength.

\[ L = \frac{1}{2}\lambda_1 \quad \implies \quad \lambda_1 = 2L \]

• Second Harmonic (\(f_2\)): Consists of two antinodes and one node in the middle. The length \(L\) is equal to one full wavelength.

\[ L = \lambda_2 \quad \implies \quad \lambda_2 = L \]

• Third Harmonic (\(f_3\)): Consists of three antinodes.

\[ L = \frac{3}{2}\lambda_3 \quad \implies \quad \lambda_3 = \frac{2L}{3} \]

General Rule for strings (or open tubes): The possible wavelengths are given by:
\[ \lambda_n = \frac{2L}{n} \quad \text{where } n = 1, 2, 3, ... \]

B. Standing Waves in Pipes (Open at both ends)

In sound waves, an open end acts as a point of maximum movement (an Antinode). The formulas for open pipes are the same as for strings fixed at both ends.

C. Standing Waves in Pipes (Closed at one end)

A closed end restricts movement, acting as a Node. An open end is an Antinode. Therefore, the possible patterns must have a Node at one end and an Antinode at the other. The simplest pattern (\(n=1\)) contains only a quarter of a wavelength.

• First Harmonic (\(f_1\)): \(L = \frac{1}{4}\lambda_1 \quad \implies \quad \lambda_1 = 4L \)
• Third Harmonic (\(f_3\)): \(L = \frac{3}{4}\lambda_3 \quad \implies \quad \lambda_3 = \frac{4L}{3} \) (The second harmonic, \(n=2\), is impossible here.)

General Rule for closed pipes: Only odd harmonics are possible.
\[ \lambda_n = \frac{4L}{n} \quad \text{where } n = 1, 3, 5, ... \]

3. Resonance

Resonance occurs when the driving frequency applied to an oscillating system is equal to the system's natural frequency (or resonant frequency). When this happens, the system absorbs energy efficiently and oscillates with a very large amplitude.

Analogy: Pushing a child on a swing. If you push at exactly the right time (the natural frequency), the amplitude (how high they swing) gets massive very quickly.

In standing waves, resonance means exciting the medium (string, air column) at one of its harmonic frequencies, leading to a strong, stable wave pattern.

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C.5 The Doppler Effect (HL Only)

Attention SL Students: This topic is for Higher Level (HL) only. You can skip this section for your core examinations, but it's fascinating Physics!

1. Understanding the Doppler Effect

The Doppler Effect is the apparent change in the frequency (and wavelength) of a wave when the source of the wave and the observer are moving relative to each other.

The Classic Example: An ambulance siren. As the ambulance approaches, the sound waves are compressed, leading to a higher frequency (higher pitch). As it moves away, the waves are stretched out, leading to a lower frequency (lower pitch).

2. Key Concepts and Equations

Let:

• \(f\): actual frequency emitted by the source
• \(f'\): observed (apparent) frequency
• \(v\): speed of the wave in the medium (e.g., speed of sound)
• \(v_o\): speed of the observer
• \(v_s\): speed of the source

The general formulas used for the Doppler effect in sound rely on two separate cases: observer motion and source motion. Note that the sign convention is crucial!

A. Observer Motion (Source is stationary)

When the observer moves, the number of wave fronts encountered per second changes.

\[ f' = f \left( \frac{v \pm v_o}{v} \right) \]

• Use \(+ v_o\) if the observer is moving TOWARDS the source (higher frequency).
• Use \(- v_o\) if the observer is moving AWAY from the source (lower frequency).

B. Source Motion (Observer is stationary)

When the source moves, the wavelength changes (compression/stretching).

\[ f' = f \left( \frac{v}{v \mp v_s} \right) \]

• Use \(- v_s\) if the source is moving TOWARDS the observer (denominator is smaller, so \(f'\) is higher).
• Use \(+ v_s\) if the source is moving AWAY from the observer (denominator is larger, so \(f'\) is lower).

3. Applications of the Doppler Effect

A. Light Waves (Astronomy)

The Doppler effect applies equally to electromagnetic waves (light), although the relativistic formula must be used for very high speeds. For speeds much less than the speed of light (\(c\)), we use:

\[ \frac{\Delta \lambda}{\lambda} \approx \frac{v}{c} \]

• Redshift: If a light source (like a galaxy) is moving AWAY from us, the observed wavelength is stretched (shifted toward the red end of the spectrum, \(\lambda\) increases). This is evidence for the expansion of the universe.
• Blueshift: If a light source is moving TOWARDS us, the observed wavelength is compressed (shifted toward the blue end of the spectrum, \(\lambda\) decreases).

B. Radar and Medical Imaging

Police use radar guns, which emit electromagnetic waves. The Doppler shift of the reflected wave tells the gun the speed of the moving vehicle. Similarly, doctors use Doppler ultrasound to measure blood flow velocity.

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That completes the "Wave Phenomena" chapter! You now have the tools to understand how waves interact, combine, and change frequency due to motion. This is vital knowledge that bridges the gap between mechanics and the study of light and sound. Great work!